Fuglede conjectured [1]} that a bounded domain \(S \subset \mathbb {R}^d\)  tiles the \(d\) -dimensional Euclidean space if and only if the set of \(L^2(S)\)  functions admits an orthogonal basis of exponential functions. The conjecture might have been motivated by Fuglede's result [1]} that it is true when the tiling complement or the spectrum is a lattice in
\(\mathbb {R}^n\) . The conjecture was disproved in [3]}. Tao considered a discrete version of the original conjecture and constructed spectral sets in \(\mathbb {Z}_2^{11}\)  and \(\mathbb {Z}_3^5\) , which are not tiles. The latter example was lifted to \(\mathbb {R}^5\)  to refute the spectral-tile direction of Fuglede's conjecture.
Matolcsi [4]} proved that the spectral-tile direction of the conjecture
fails in \(\mathbb {R}^4\) . Kolountzakis and Matolcsi
[5]}, [6]} and Farkas, Matolcsi and Móra [7]}
provided counterexamples in \(\mathbb {R}^3\)  for both
direction of the conjecture. The case of elementary abelian groups is
